Hadronic contributions to $\alpha (Q^2)$ and ${\sin }^2{\theta }_W(Q^2)$ from spectral reconstruction of lattice-QCD data

We present preliminary results from a lattice-QCD study of the hadronic contributions to the running of the electromagnetic coupling, $\mathrm{\Delta }\alpha (Q^2)$, and the electroweak mixing angle, $\mathrm{\Delta }{\sin }^2{\theta }_W(Q^2)$. Using $N_f=2+1+1$ HISQ ensembles at physical quark masses, we discuss the challenges posed by strong statistical correlations in the time-momentum representation and propose a spectral-reconstruction strategy to obtain controlled continuum-extrapolated results across the full energy range.